Stat 5101 Lecture Notes - School of Statistics

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48 Stat 5101 (Geyer) Course NotesCorollary 2.20. If W , X, Y , and Z are random variables having first andsecond moments and a, b, c, and d are constants, thencov (aW + bX, cY + dZ)=ac cov(W, Y )+ad cov(W, Z)+ bc cov(X, Y )+bd cov(X, Z) (2.26)var (aX + bY )=a 2 var(X)+2ab cov(X, Y )+b 2 var(Y ) (2.27)cov (W + X, Y + Z) =cov(W, Y )+cov(W, Z)+cov(X, Y )+cov(X, Z) (2.28)var (X + Y )=var(X)+2cov(X, Y ) + var(Y ) (2.29)No proof is necessary, since all of these equations are special cases of thosein Theorem 2.16 and its corollaries.This section contains a tremendous amount of “equation smearing.” It is thesort of thing for which the acronym MEGO (my eyes glaze over) was invented.To help you remember the main point, let us put Corollary 2.19 in words.The variance of a sum is the sum of the variances plus the sum oftwice the covariances.Contrast this with the much simpler slogan about expectations on p. 35.The extra complexity of the of the variance of a sum contrasted to theexpectation of a sum is rather annoying. We would like it to be simpler. Unfortunatelyit isn’t. However, as elsewhere in mathematics, what cannot beachieved by proof can be achieved by definition. We just make a definition thatdescribes the nice case.Definition 2.4.1.Random variables X and Y are uncorrelated if cov(X, Y )=0.We also say a set X 1 , ..., X n of random variables are uncorrelated if eachpair is uncorrelated. The reason for the name “uncorrelated” will become clearwhen we define correlation.When a set of random variables are uncorrelated, then there are no covarianceterms in the formula for the variance of their sum; all are zero by definition.Corollary 2.21. If the random variables X 1 , ..., X n are uncorrelated, thenIn words,var(X 1 + ...+X n ) = var(X 1 )+...+ var(X n ).The variance of a sum is the sum of the variances if (big if) thevariables are uncorrelated.Don’t make the mistake of using this corollary or the following slogan whenits condition doesn’t hold. When the variables are correlated (have nonzerocovariances), the corollary is false and you must use the more general formulaof Corollary 2.19 or its various rephrasings.
2.4. MOMENTS 49What happens to Corollary 2.17 when the variables are uncorrelated is leftas an exercise (Problem 2-16).At this point the reader may have forgotten that nothing in this section hasyet been proved, because we deferred the proof of Theorem 2.16, from whicheverything else in the section was derived. It is now time to return to that proof.Proof of Theorem 2.16. First defineU =V =m∑a i X ii=1n∑b j Y jj=1Then note that by linearity of expectationThenµ U =µ V =m∑a i µ Xii=1n∑b j µ Yjj=1cov(U, V )=E{(U−µ U )(V − µ V )}⎧( ⎨ m)∑m∑⎛ ⎞⎫n∑n∑ ⎬= E a⎩ i X i − a i µ Xi⎝ b j Y j − b j µ Yj⎠⎭i=1i=1j=1 j=1⎧⎫⎨ m∑n∑ ( ) ⎬= E (a i X i − a i µ Xi ) bj Y j − b j µ Yj⎩⎭i=1j=1⎧⎫⎨ m∑ n∑⎬= E a i b j (X i − µ Xi )(Y j − µ Yj )⎩⎭i=1 j=1m∑ n∑= a i b j E { (X i − µ Xi )(Y j − µ Yj ) }=i=1 j=1m∑i=1 j=1n∑a i b j cov(X i ,Y j ),the last equality being the definition of covariance, the next to last linearityof expectation, and the rest being just algebra. And this proves the theorembecause cov(U, V ) is the left hand side of (2.21) in different notation.
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Stat 5101 Lecture NotesCharles J. G
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Contents1 Random Variables and Chan
Page 5 and 6: CONTENTSv5.1.4 Variance Matrices ..
Page 7 and 8: CONTENTSviiD Relations Among Brand
Page 9 and 10: Chapter 1Random Variables andChange
Page 11 and 12: 1.1. RANDOM VARIABLES 3Sometimes it
Page 13 and 14: 1.1. RANDOM VARIABLES 5that is, X i
Page 15 and 16: 1.2. CHANGE OF VARIABLES 7the union
Page 17 and 18: 1.2. CHANGE OF VARIABLES 9Thus even
Page 19 and 20: 1.2. CHANGE OF VARIABLES 11Every in
Page 21 and 22: 1.2. CHANGE OF VARIABLES 13where f
Page 23 and 24: 1.3. RANDOM VECTORS 151.3.1 Discret
Page 25 and 26: 1.4. THE SUPPORT OF A RANDOM VARIAB
Page 27 and 28: 1.5. JOINT AND MARGINAL DISTRIBUTIO
Page 29 and 30: 1.5. JOINT AND MARGINAL DISTRIBUTIO
Page 31 and 32: 1.6. MULTIVARIABLE CHANGE OF VARIAB
Page 39 and 40: Chapter 2Expectation2.1 Introductio
Page 41 and 42: 2.3. BASIC PROPERTIES 33expectation
Page 43 and 44: 2.3. BASIC PROPERTIES 35that is, th
Page 45 and 46: 2.4. MOMENTS 37The Multiplicativity
Page 47 and 48: 2.4. MOMENTS 39holds for all real-v
Page 49 and 50: 2.4. MOMENTS 41simple, it is often
Page 51 and 52: 2.4. MOMENTS 43In contrast, for all
Page 53 and 54: 2.4. MOMENTS 45is the sum of the ex
Page 55: 2.4. MOMENTS 47Proof. Just take a i
Page 59 and 60: 2.4. MOMENTS 51This inequality is a
Page 61 and 62: 2.4. MOMENTS 53(a)Why does (2.7) as
Page 63 and 64: 2.5. PROBABILITY THEORY AS LINEAR A
Page 83 and 84: 2.6. PROBABILITY IS A SPECIAL CASE
Page 85 and 86: 2.7. INDEPENDENCE 772.7 Independenc
Page 87 and 88: 2.7. INDEPENDENCE 79Then X and Y ar
Page 89 and 90: 2.7. INDEPENDENCE 81Show that the f
Page 91 and 92: Chapter 3Conditional Probability an
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Page 95 and 96: 3.2. CONDITIONAL PROBABILITY DISTRI
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Page 103 and 104: 3.4. JOINT, CONDITIONAL, AND MARGIN
Page 105 and 106: 3.4. JOINT, CONDITIONAL, AND MARGIN
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3.4. JOINT, CONDITIONAL, AND MARGIN
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3.5. CONDITIONAL EXPECTATION AND PR
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Chapter 4Parametric Families ofDist
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4.1. LOCATION-SCALE FAMILIES 113The
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4.2. THE GAMMA DISTRIBUTION 115Show
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4.3. THE BETA DISTRIBUTION 117cours
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4.4. THE POISSON PROCESS 119Hence t
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4.4. THE POISSON PROCESS 121Definit
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4.4. THE POISSON PROCESS 123measure
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4.4. THE POISSON PROCESS 1254-3. A
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Chapter 5Multivariate DistributionT
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5.1. RANDOM VECTORS 129Again like r
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5.1. RANDOM VECTORS 1315.1.6 Covari
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5.1. RANDOM VECTORS 1335.1.7 Linear
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5.1. RANDOM VECTORS 135Example 5.1.
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5.1. RANDOM VECTORS 137Example 5.1.
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5.1. RANDOM VECTORS 139where we hav
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5.2. THE MULTIVARIATE NORMAL DISTRI
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5.3. BERNOULLI RANDOM VECTORS 151Th
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5.3. BERNOULLI RANDOM VECTORS 153yo
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5.4. THE MULTINOMIAL DISTRIBUTION 1
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Chapter 6Convergence Concepts6.1 Un
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6.1. UNIVARIATE THEORY 167It simpli
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6.1. UNIVARIATE THEORY 169density o
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6.1. UNIVARIATE THEORY 171Rewriting
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6.1. UNIVARIATE THEORY 173Comment T
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6.1. UNIVARIATE THEORY 175Problems6
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Chapter 7Sampling Theory7.1 Empiric
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7.1. EMPIRICAL DISTRIBUTIONS 1797.1
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7.1. EMPIRICAL DISTRIBUTIONS 181Def
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7.1. EMPIRICAL DISTRIBUTIONS 183To
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7.2. SAMPLES AND POPULATIONS 185The
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7.2. SAMPLES AND POPULATIONS 187•
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7.3. SAMPLING DISTRIBUTIONS OF SAMP
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Appendix AGreek LettersTable A.1: T
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Appendix BSummary of Brand-NameDist
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B.1. DISCRETE DISTRIBUTIONS 215B.1.
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B.2. CONTINUOUS DISTRIBUTIONS 217Th
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B.2. CONTINUOUS DISTRIBUTIONS 219B.
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B.4. DISCRETE MULTIVARIATE DISTRIBU
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B.5. CONTINUOUS MULTIVARIATE DISTRI
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B.5. CONTINUOUS MULTIVARIATE DISTRI
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Appendix CAddition Rules forDistrib
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Appendix DRelations Among BrandName
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Appendix EEigenvalues andEigenvecto
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E.2. EIGENVALUES AND EIGENVECTORS 2
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E.2. EIGENVALUES AND EIGENVECTORS 2
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E.3. POSITIVE DEFINITE MATRICES 237
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Appendix FNormal Approximations for
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